Barnes, Bruce A. The properties *-regularity and uniqueness of \(C^*\)-norm in a general *-algebra. (English) Zbl 0532.46031 Trans. Am. Math. Soc. 279, 841-859 (1983). In this paper two important properties of a Banach *-algebra A are studied: the existence of a unique \(C^*\)-norm on A and the \({}^*\)- regularity of A. These properties are involved with the representation theory of A. For example when A is a reduced Banach \({}^*\)-algebra, then A has a unique \(C^*\)-norm if and only if every separating collection of \({}^*\)-representations of A lifts to a separating collection of \({}^*representations\) of the \(C^*\)-enveloping algebra of A. A number of basic results are derived concerning these important properties, and some applications are given in the case where \(A=L^ 1(G)\). Cited in 2 ReviewsCited in 16 Documents MSC: 46K10 Representations of topological algebras with involution 46H05 General theory of topological algebras 46L05 General theory of \(C^*\)-algebras Keywords:Banach *-algebra; unique \(C^*\)-norm; \({}^*\)-regularity; representation theory; reduced Banach \({}^*\)-algebra,; separating collection of \({}^*\)-representations PDFBibTeX XMLCite \textit{B. A. Barnes}, Trans. Am. Math. Soc. 279, 841--859 (1983; Zbl 0532.46031) Full Text: DOI